US8539012B2 - Multi-rate implementation without high-pass filter - Google Patents
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- US8539012B2 US8539012B2 US13/006,164 US201113006164A US8539012B2 US 8539012 B2 US8539012 B2 US 8539012B2 US 201113006164 A US201113006164 A US 201113006164A US 8539012 B2 US8539012 B2 US 8539012B2
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
- H03H17/00—Networks using digital techniques
- H03H17/02—Frequency selective networks
- H03H17/06—Non-recursive filters
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
- H03H17/00—Networks using digital techniques
- H03H17/02—Frequency selective networks
- H03H17/0223—Computation saving measures; Accelerating measures
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- H—ELECTRICITY
- H03—ELECTRONIC CIRCUITRY
- H03H—IMPEDANCE NETWORKS, e.g. RESONANT CIRCUITS; RESONATORS
- H03H17/00—Networks using digital techniques
- H03H17/02—Frequency selective networks
- H03H17/0283—Filters characterised by the filter structure
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- the field of the invention is digital signal filtering and in particular methods to improve low frequency resolution without increasing processing requirements.
- Broadband digital signals often require filtering for their intended use.
- signals may be filtered (often referred to as equalizing) to compensate for characteristics of speakers and the listening environment.
- Such digital signals must be at a sufficiently high sample rate to carry the high frequency signal components.
- Finite Impulse Response (FIR) (or transversal) filters are preferred in many applications to maintain linear phase or minimum phase for accurate sound reproduction.
- FIR Finite Impulse Response
- the filtering In order to filter such signals using a single FIR filter, the filtering must be performed at the high sample rate of the digital signal, and to achieve high resolution filtering for low frequencies present in the signal, a very long FIR filter is also required. In some audio systems the resulting processing requirements cannot be performed economically.
- Multi-rate filters have been introduced to overcome the very long FIR filter requirement of single filter implementations.
- Such multi-rate filters separate the digital signal into at least two bands.
- a high frequency signal band is processed at the high sample rate of the original signal using a short FIR filter and a parallel low frequency band is first down sampled to a lower sample rate, filtered using a FIR filter which may be much shorter than the traditional filter required by the single filter implementation, and up sampled back to the original sample rate.
- the two filtered signals are then summed to provide the desired filtered signal.
- Such multi-rate filters include a high pass filter to provide the high frequency signal, and a separate low pass filter to provide the separate parallel low frequency signal.
- IIR Infinite Impulse Response
- low pass filters might be used in the multi-rate filter, but unfortunately such IIR filters introduce differing group delay above and below the transition frequency and may distort the resulting audio signal.
- linear phase FIR's may be used, but unfortunately the FIR high and low pass filters are computationally intense, and minimize the benefit of the multi-rate filtering approach.
- the present invention addresses the above and other needs by providing a filtering method which approximates a target Finite Impulse Response (FIR) (or transversal) filter and reduces computational requirements by eliminating high pass filtering required by known multi-rate filters.
- An input signal is copied into two identical signals and processed in parallel by a full-rate path, and by a reduced-rate path.
- Parallel filters are computed and applied in each path, the reduced-rate signal is up-sampled, and the two signals summed.
- the high pass filter required by known multi-rate filters is eliminated and the low pass filter in the prior art is implicit in a down sampling.
- Linear phase FIR filters are used for down and up sampling, resulting in constant group delay. Added benefits include the option of zero added latency through the filtering and the constant group delay added to the target FIR filter.
- the user may choose criteria such as minimum resolution in each band.
- a method of calculating a pair of FIR filters, and a corresponding signal processing diagram for applying the calculated filters is provided.
- the filters are derived to eliminate the need for a separate high pass filter required by known multi-rate filters.
- a signal filtering method which includes an option of zero added latency through the filtering and constant group delay added to the target FIR.
- the user may choose criteria such as minimum resolution in each band for an approximation of the target FIR filter.
- FIG. 1 is a prior art digital filter with a single Finite Impulse Response (FIR) filter applied to the input signal.
- FIR Finite Impulse Response
- FIG. 2 is a prior art multi-rate digital filter with two parallel FIR filters applied to the input signal.
- FIG. 3 is an improved multi-rate filter according to the present invention applied to the input signal.
- FIG. 4 is a method for computing a pair of FIR filters according to the present invention for use in the improved multi-rate filter.
- FIG. 5A is a target impulse response.
- FIG. 5B is a target frequency response.
- FIG. 6 is a window function.
- FIG. 7A is an impulse response of a full-rate filter according to the present invention.
- FIG. 7B is a frequency response of the full-rate filter according to the present invention.
- FIG. 8A is an impulse response of a residual filter according to the present invention.
- FIG. 8B is a frequency response of the residual filter according to the present invention.
- FIG. 9A is an impulse response of an anti-aliasing filter according to the present invention.
- FIG. 9B is a frequency response of the anti-aliasing filter according to the present invention.
- FIG. 10A is an impulse response of a reduced-rate filter according to the present invention.
- FIG. 10B is a frequency response of the reduced-rate filter according to the present invention.
- FIG. 11A is an impulse response of an up-samples signal from the reduced-rate filter according to the present invention.
- FIG. 11B is a frequency response of an up-samples signal from the reduced-rate filter according to the present invention.
- FIG. 12A is an impulse response of the summed signal according to the present invention.
- FIG. 12B is an impulse response of the target FIR filter for comparison to the impulse response of the summed signal.
- FIG. 13A is a frequency response of the summed signal according to the present invention.
- FIG. 13B is a frequency response of the target FIR filter for comparison to the frequency response of the summed signal.
- FIG. 14 shows an embodiment of the an improved multi-rate filter according to the present invention including a down sampled and reversed version of the anti-aliasing filter in the reduced-rate path.
- the Target FIR filter 10 must perform operations at the high sample rate of the input signal 12 and also be of sufficient length to provide accurate filtering to low frequency components in the input signal. The resulting processing may require a very large number of operations at the high sample rate.
- FIR Finite Impulse Response
- FIG. 2 A prior art multi-rate digital filter 20 with two parallel FIR filters, FIR H 24 and FIR L 34 , applied to the input signal 12 is shown in FIG. 2 .
- the input signal 12 is processed by a high pass filter 22 providing a high frequency signal 23 for processing by the FIR H 24 to provide a filtered high frequency signal 25 .
- the input signal 12 is also processed by a low pass filter 30 to produce a low frequency signal 31 at the original sample rate, down sampled by a down-sampling filter 32 to produce a down sampled low frequency signal 33 .
- the down sampled low frequency signal 33 is processed by the FIR L 34 to provide a filtered down sampled low frequency signal 35 which is up-sampled by up-sampler 36 to provide a filtered low frequency signal 37 at the original sample rate.
- the filtered high frequency signal 25 and the filtered low frequency signal 37 are summed by the summer 26 to provide the filtered signal 14 .
- the multi-rate digital filter 20 reduces the processing required by the Target FIR filter 10 because the FIR L filter 34 operates at a much lower sample rate than the Target FIR filter 10 .
- the low pass filter 30 may be an anti-aliasing filter intrinsic in the down-sampler 32 .
- the multi-rate filter 40 includes a second high frequency (or full-rate) filter FIR F 42 replacing both the high pass filter 22 and the filter FIR H 24 and a second low frequency (or reduced-rate) filter FIR R 46 replacing both the low pass filter 30 and the filter FIR L 34 .
- the full-rate filter FIR F 42 filters the input signal 12 to provide a filtered full-rate signal 43 .
- a second down-sampler 44 down samples the input signal 12 to provide a down sampled (or reduced-rate) signal 45 .
- the reduced-rate filter FIR R 46 filters the reduced-rate signal 45 to provide a filtered reduced-rate signal 47 .
- a second up sampler 48 up samples the filtered reduced-rate signal 47 to provide an up sampled filtered signal 49 .
- the summer 26 sums the filtered full-rate signal 43 with the up sampled filtered signal 49 to provide the filtered signal 14 .
- the resulting filter 40 thus reduces overall computational requirements by eliminating high pass filter 22 in the known multi-rate filter 20 .
- the resulting filter 40 provides added benefits including the option of zero added latency through the filtering and constant group delay added to the target FIR because the IIR high and low pass filters are no longer included.
- FIG. 4 is a method for computing the full-rate FIR F and reduced-rate FIR R FIR filters 42 and 46 according to the present invention for use in the improved multi-rate filter 40 .
- the filters are computed using the following steps. Choosing a target FIR filter to be modeled at step 100 . (A minimum, maximum, or linear phase filter is preferred. In order to achieve zero added latency, a minimum phase filter is preferred.) Choosing a decimation factor and designing an appropriate anti-aliasing filter A at step 102 . (The anti-aliasing filter A is preferably linear phase and has an order which is an integer multiple of the decimation factor.) Choosing a length M of a full-rate filter FIR F at step 104 .
- the length of the full-rate filter FIR H must be at least 1.5 times the length of the anti-aliasing filter A. Selecting a window W of length M used to create the full-rate filter FIR F at step 106 .
- the window should start with a flat (1.0) region that has length in samples 1.5* ⁇ order of the anti-aliasing filter A>. If different filters are used for anti-aliasing the signal, anti-aliasing the residual filter R, and anti-imaging, then the 1.0 region should be the total group delay of all anti-aliasing and anti-imaging filters.
- the right half of a typical window e.g.
- a Hann window should be calculated to taper from 1 down to 0 at the length of the full-rate filter FIR F .
- an anti-aliasing filter is present as part of the down-sampler 44 and applied in a single step with the down sampling.
- the up-sampler 48 includes “reconstruction or “anti-imaging” filtering. The up sampling and reconstruction filter are also preferably applied in a single step, since only 1/n samples of the reconstruction filter input are non-zero.
- Various up-samplers are known in the art and suitable for use with the present invention.
- the high rate filter FIR F is symmetrically windowed, resulting in a reduced-rate filter FIR R with zeros in the center.
- An efficient implementation of the reduced-rate filter FIR R is obtained by breaking the reduced-rate filter FIR L into two non-zero parts and applying in a delay (or gap), corresponding to the number of zero sample discarded, to signal samples between the first non-zero part and the second non-zero part of the reduced-rate filter FIR R .
- FIGS. 5A-13B A target impulse response of the prior art target FIR filter 10 is shown in FIG. 5A and a target frequency response of the prior art target FIR filter 10 is shown in FIG. 5B .
- the goal of the present invention is to obtain a good approximation of the target impulse and frequency responses with reduced processing requirements.
- the exemplar target filter in FIGS. 5A and 5B has a 48 k Hz sample rate and a filter length N of 1024.
- a window function W for windowing the target FIR filter 10 to obtain the full-rate filter FIR F 42 is shown in FIG. 6 .
- the window function W is used to reduce the length of the target FIR filter 10 to the desired full-rate FIR filter length. If zero latency is desired, the window function W must begin with a region of 1.0 values that is equal in length to the total latency of the reduced-rate path. In the example of FIG. 6 , the window function W has a length M of 256, and the first 128 elements have values of 1.0. This latency is the sum of the group delays of the filter used to decimate the signal, the filter used to decimate the FIR, and the filter used to interpolate the filtered signal.
- the phase of the down-sampler 44 see FIG.
- the window function W must be long enough to avoid an abrupt truncation of the full-rate filter FIR F 42 .
- One of several known window functions W may be used, and an example of a suitable window function W comprises a left (leading) half of all ones and a right (trailing half), the right half of a known window.
- suitable known windows include a Hann window, a rectangular window, a hamming window, a tukey window, a cosine window, a lanczos window, a Bartlett window, a triangular window, a Gaussian window, a Bartlett-Hann window, a Blackman window, a Kaiser window, a Nuttall window, a Blackman-Harris, Blackman-Nuttall window, a Flat top window, a Bessel window, or a similar window.
- a preferred window is the Hann window, the Gaussian window, and the Kaiser window.
- the length M of the window function W is preferably selected based on desired properties of the full-rate path and reduced-rate path. Zero latency is achieved by including 1.0's to compensate for the group delay of the reduced-rate path, but this requires a trade-off in the transition region of the window function W (the path from 1.0 s to approximately 0 at the end, i.e., the right half of the window function W).
- the overall length of the window function W is determined by the desired length of the full-rate filter, or vice-versa.
- the overall length M of the window function W, the number of 1.0 s, and the shape of the right half of the window function W, may all be adjusted to trade off characteristics, e.g., windowing over too short a region will induce artifacts comparable to truncation (i.e., rectangular windowing).
- the window function W is applied to the target FIR filter 10 by term by term multiplying each sample of the target FIR filter 10 by the corresponding term of the window function W up to the length M of the window.
- the length M of the window W is preferably selected based on parameters of the reduced-rate path. Zero added latency is accomplished by setting the number of 1.0 elements (i.e., M/2) of the window W to the total group delay of the reduced-rate path's decimation and interpolation filters (not necessarily the same filter), as well as the residual FIR anti-aliasing filter A (which may or may not be the same as the signal decimation/interpolation filter(s)).
- the latency may be adjusted as desired, but must be compensated for by a delay in either the full-rate path or the reduced-rate path if the total group delay of the reduced-rate path is made more or less than the full-rate path.
- a delay in either the full-rate path or the reduced-rate path if the total group delay of the reduced-rate path is made more or less than the full-rate path.
- there are no 1.0's in the window and the entire reduced-rate path latency must be taken into account by applying a delay to the full-rate path before the summation.
- FIG. 7A An impulse response of the full-rate filter FIR F 42 is shown in FIG. 7A and a frequency response of the full-rate filter FIR F 42 is shown in FIG. 7B .
- the resulting full-rate filter FIR F 42 is applied to the input signal 12 at the full sampling rate.
- the frequency response appears to have a high-pass effect
- the actual response of the full-rate filter FIR F 42 will depend on the group delay of the target FIR filter 10 .
- the frequency response of the full-rate filter FIR F 42 will resemble a smoothed version of the target FIR filter 10 .
- the residual filter R is obtained by subtracting (term by term) the windowed full-rate filter FIR F 42 having a length M from the first M elements of target FIR filter 10 having a length N to obtain R having a length of N:
- the impulse response of the N element residual filter R is shown in FIG. 8A and the frequency response of the residual filter R is shown in FIG. 8B .
- the frequency response has the appearance of a low-pass filter even though no low-pass filter has been applied. This appearance is because most of the high frequency energy occurs early in the minimum phase response, and is present in the full-rate FIR filter FIR F 42 .
- R′ ( r M/2+1 . . . r N )
- Proper sizing of the leading 1.0 flat region of the window function W allows the reduced-rate path response to be summed with the full-rate path response without a need to apply delay to the full-rate path.
- the length of the flat 1.0 leading region of half the length of the window W is an approximate preferred length.
- the actual length of the 1.0 leading region may vary depending on applications and multi-rate filters computed using a window W with the 1.0 leading region greater than or less than half the window W length are intended to come within the scope of the present invention.
- the down sampler 44 includes an anti-aliasing filter A followed by down sampling.
- An impulse response of a suitable anti-aliasing filter A of length J (where J is 85 in this example) according to the present invention is shown in FIG. 9A and a frequency response of the anti-aliasing filter A is shown in FIG. 9B .
- the anti-aliasing filter A is designed to control aliasing resulting from the following down sampling and from the down-sampling to obtain the reduced-rate filter 46 .
- the anti-aliasing filter A may also be used to down sample and decimate the input signal 12 in the down sampler 44 (see FIG. 4 ), when appropriate.
- FIG. 10A An impulse response of the reduced-rate filter 46 according to the present invention is shown in FIG. 10A and a frequency response of the reduced-rate filter 46 is shown in FIG. 10B .
- the reduced-rate filter 46 is computed by convolving the anti-aliasing filter A with the residual filter R′, then down-sampling (discarding n ⁇ 1 out of every n samples, where n equals 4 in this example). The frequency response can be seen to roll off in the high frequencies due to the reduced-rate filter FIR R .
- the anti-aliasing filter A comprises the elements d
- FIG. 11 a An impulse response of the up-sampled signal 49 (see FIG. 3 ) from the reduced-rate filter path according to the present invention is shown in FIG. 11 a and a frequency response of an up-sampled signal 49 is shown in FIG. 11B .
- the reduced-rate filter FIR R has 256 elements corresponding to a down sampling of four.
- FIG. 12A An impulse response of the summed signal 14 (see FIG. 30) according to the present invention is shown in FIG. 12A and a corresponding plot of the impulse response of the target FIR filter 10 , for comparison to the impulse response of the summed signal 14 , is shown in FIG. 12B .
- a frequency response of the summed signal 14 is shown in FIG. 13A and a corresponding plot of the frequency response of the target FIR filter 10 , for comparison to the frequency response of the summed signal 14 , is shown in FIG. 13B .
- the filter according to the present invention has nearly the same impulse response as the target FIR filter, with slight smoothing beginning in the windowed region and continuing to the end of the filter.
- the filter according to the present invention has a longer impulse response due to the decimation and interpolation filters applied, but, in this example, no additional processing delay is incurred.
- the present invention thus provides virtually the same response as the target FIR filter with slight smoothing in the high frequencies, but achieved with reduced processing requirements.
- the present invention is described above for cases where the target filter is efficiently performed using a full-rate path and a single reduced-rate path.
- the single reduced-rate path shown in FIG. 4 may still require significant processing.
- the methods described above may be applied a second (or more) time in a nested topology replacing the reduced-rate filter FIR R 46 with additional filters of the form of the filter 40 .
- the filter may include three nested reduced-rate paths.
- the resulting down sampled (every n ⁇ 1 out of n samples are removed) filter is always asymmetric.
- any even anti-aliasing filter A when decimated by an even factor, n exhibits asymmetry.
- a resulting asymmetric impulse response is not linear phase and does not have constant delay for all frequencies.
- any multi-rate processing where the reduced-rate signal must be able to maintain the same phase as the full-rate signal (whether high-pass filtered or not), this would make even length FIR filters unacceptable for use as anti-aliasing filters.
- Such convolution may be applied to achieve constant group delay through the reduced-rate path of a multi-rate filter by including a down sampled and reversed filter RV 52 version of the anti-aliasing filter A in the reduced-rate signal path after the down-sampler as shown in FIG. 14 .
- the reverse filter RV is computed by down sampling and reversing the anti-aliasing filter A. For example, where the anti-aliasing filter A is an 86 element filter, the reverse filter RV is a 21 element filter after down sampling by a factor of four.
- the reverse filter RV may effectively be implemented by convolving the anti-aliasing filter A with a full-rate filter (for example, the residual filter R) and down sampling the result, and further incorporating a different phase (or offset) in the down-sampling process. Rather than keeping the first sample, and discarding the following n ⁇ 1 samples leaving (a 1 , a n+1 , a 2n+1 , . . .
- an offset OS in samples
- the retained samples become (a 1+OS , a n+1+OS , a 2n+1+OS , . . . )
- Doing this with the correct “phase” (or offset) has the same effect as reversing the down-sampled decimation filter because the original filter was symmetrical.
- the same anti-aliasing filter A may also be used in the down sampler 44 (see FIG. 3 ).
Abstract
Description
FIR=(c1,c2,c2, . . . ,c N)
W=(w1,w2,w3, . . . w M)
FIRF=(c 1 w 1 ,c 2 w 2 ,c 3 w 3 , . . . c M w M)
R=FIR−FIRF=(c1−c1w1,c2−c2w2, . . . c M −c M w M ,c M+1 , . . . c N)
R=(0,0,0 . . . 0,r M/2+1 . . . r N)
R′=(r M/2+1 . . . r N)
A=(a 1 ,a 2 ,a 3 . . . a j)
FIRg(n)=Σj-1 j-J A(j)R′(n−f)
896+85−1=980
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